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   "source": [
    "# 高等数学框架\n",
    "\n"
   ]
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    "于是\n",
    "\n",
    "$ W = \\sum_{i=1}^n \\Delta W_i \\approx \\sum_{i=1}^n \\left[ P(\\xi_i, \\eta_i) \\Delta x_i + Q(\\xi_i, \\eta_i) \\Delta y_i \\right] $\n",
    "\n",
    "用 $\\lambda$表示 $n$ 个小弧段的最大长度，令 $\\lambda \\rightarrow 0$ 取上述和的极限，所得到的极限自然地被认作变力 $F$ 沿有向曲线弧所作的功，即\n",
    "\n",
    "$ W = \\lim_{\\lambda \\rightarrow 0} \\sum_{i=1}^n \\left[ P(\\xi_i, \\eta_i) \\Delta x_i + Q(\\xi_i, \\eta_i) \\Delta y_i \\right] $\n",
    "\n",
    "这种和的极限在研究其他问题时也会遇到。现在引进下面的定义：\n",
    "\n",
    "定义 设 L 为 xOy 面内从点 A 到点 B的一条有向光滑曲线弧，函数 P(x, y)与 Q(x, y)在 L 上有界。在 L 上沿L 的方向任意插入一点列 $M_1(x_1, y_1)$, $M_2(x_2, y_2), \\cdots, M_{n-1}(x_{n-1}, y_{n-1})$，把 \\(L\\) 分成 \\(n\\) 个有向小弧段\n",
    "\n",
    "$\\widehat{M_{i-1} M_i} \\quad (i=1,2,\\cdots, n; M_0=A, M_n=B) $\n",
    "\n",
    "设 $\\Delta x_i = x_i - x_{i-1}, \\Delta y_i = y_i - y_{i-1}$，点 $(\\xi_i, \\eta_i)$ 为 $\\widehat{M_{i-1} M_i}$ 上任意取定的点，作乘积 $P(\\xi_i, \\eta_i) \\Delta x_i (i=1,2,\\cdots, n)$，并作和 $\\sum_{i=1}^n P(\\xi_i, \\eta_i) \\Delta x_i$，如果当各小弧段长度的最大值 $\\lambda \\rightarrow 0$ 时，这和的极限总存在，且与曲线弧 L 的分法及点 $(\\xi_i, \\eta_i)$ 的取法无关，那么称此极限为函数 P(x, y) 在有向曲线弧 L 上对坐标 x 的曲线积分，记作 $\\int_L P(x, y) dx$。类似地，如果 $\\lim_{\\lambda \\rightarrow 0} \\sum_{i=1}^n Q(\\xi_i, \\eta_i) \\Delta y_i$ 总存在，且与曲线弧 L 的分法及点 $(\\xi_i, \\eta_i)$ 的取法无关，那么称此极限为函数 Q(x, y) 在有向曲线弧 L 上对坐标 y 的曲线积分，记作 $\\int_L Q(x, y) dy$。即\n",
    "\n",
    "\\[\n",
    "\\begin{align*}\n",
    "&\\int_L P(x, y) dx = \\lim_{\\lambda \\rightarrow 0} \\sum_{i=1}^n P(\\xi_i, \\eta_i) \\Delta x_i, \\\\\n",
    "&\\int_L Q(x, y) dy = \\lim_{\\lambda \\rightarrow 0} \\sum_{i=1}^n Q(\\xi_i, \\eta_i) \\Delta y_i,\n",
    "\\end{align*}\n",
    "\\]\n",
    "\n",
    "其中 P(x, y)、Q(x, y) 叫做被积函数，L 叫做积分弧段。\n",
    "\n",
    "以上两个积分也称为第二类曲线积分。\n",
    "\n",
    "在第二目中我们将看到，当 P(x, y) 与 Q(x, y) 在有向光滑曲线弧 L 上连续时，对坐标的曲线积分 $\\int_L P(x, y) dx$ 及 $\\int_L Q(x, y) dy$ 都存在。以后我们总假定 P(x, y) 与 Q(x, y) 在 L 上连续。\n",
    "\n",
    "上述定义可以类似地推广到积分弧段为空间有向曲线弧 $\\Gamma$ 的情形：\n",
    "\n",
    "$ \\int_\\Gamma P(x, y, z) dx = \\lim_{\\lambda \\rightarrow 0} \\sum_{i=1}^n P(\\xi_i, \\eta_i, \\zeta_i) \\Delta x_i $"
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    "\\[\n",
    "\\begin{align*}\n",
    "\\int_r Q(x, y, z) dy &= \\lim_{\\lambda \\rightarrow 0} \\sum_{i=1}^n Q(\\xi_i, \\eta_i, \\zeta_i) \\Delta y_i, \\\\\n",
    "\\int_r R(x, y, z) dz &= \\lim_{\\lambda \\rightarrow 0} \\sum_{i=1}^n R(\\xi_i, \\eta_i, \\zeta_i) \\Delta z_i.\n",
    "\\end{align*}\n",
    "\\]\n",
    "\n",
    "应用上经常出现的是\n",
    "\n",
    "$\n",
    "\\int_L P(x, y) dx + \\int_L Q(x, y) dy\n",
    "$\n",
    "\n",
    "这种合并起来的形式，为简便起见，把上式写成\n",
    "\n",
    "$\n",
    "\\int_L P(x, y) dx + Q(x, y) dy\n",
    "$\n",
    "\n",
    "也可写成向量形式\n",
    "\n",
    "$\n",
    "\\int_L F(x, y) \\cdot dr,\n",
    "$\n",
    "\n",
    "其中 F(x, y) = P(x, y) i + Q(x, y) j 为向量值函数，$dr = dx i + dy j$。\n",
    "\n",
    "例如，本目开始时讨论过的变力 F 所作的功可以表达成\n",
    "\n",
    "$\n",
    "W = \\int_L P(x, y) dx + Q(x, y) dy\n",
    "$\n",
    "\n",
    "或\n",
    "\n",
    "$\n",
    "W = \\int_L F(x, y) \\cdot dr\n",
    "$\n",
    "\n",
    "类似地，把\n",
    "\n",
    "$\n",
    "\\int_r P(x, y, z) dx + \\int_r Q(x, y, z) dy + \\int_r R(x, y, z) dz\n",
    "$\n",
    "\n",
    "简写成\n",
    "\n",
    "$\n",
    "\\int_r P(x, y, z) dx + Q(x, y, z) dy + R(x, y, z) dz\n",
    "$\n",
    "\n",
    "或\n",
    "\n",
    "$\n",
    "\\int_{\\Gamma} A(x, y, z) \\cdot dr,\n",
    "$\n",
    "\n",
    "其中 $A(x, y, z) = P(x, y, z) i + Q(x, y, z) j + R(x, y, z) k$，$dr = dx i + dy j + dz k$。\n",
    "\n",
    "如果 L（或 $\\Gamma$）是分段光滑的，我们规定函数在有向曲线弧 L（或 $\\Gamma$）上对坐标的曲线积分等于在光滑的各段上对坐标的曲线积分之和。\n",
    "\n",
    "根据上述曲线积分的定义，可以导出对坐标的曲线积分的一些性质。为了表达简便起见，我们用向量形式表达，并假定其中的向量值函数在曲线 L 上连续①。"
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